EE 582 (Physical Design Automation of VLSI Circuits and Systems)

Assignment 7

1. Partial differential equations

1-1. x, y, and z are variables. Find the partial derivatives of the following functions with respect to x, y, and z.

• F (x, y, z) = x² + y² + z²

• G (x, y, z) = 4x⁴ + 2x³y²z + x²z²

• H (x, y, z) = 2(x³ + 2x²y³z²)³

• J (x, y, z) = sin(x + 2y³ z⁴)

1-2. x, y, and z are variables. Find closed-form expressions for the following functions. Notice that the constants of integration are not just constants but functions of some variables.

• F (x, y, z) = ∫ (x^2 + 3 y²z) dx

• G (x, y, z) = ∫ (x cos(xy+z)) dy

2. Analytical placement

2-1. Two pins, p₁ and p₂ are located at a₁ and a₂, respectively, along the x-axis (a₁ < a₂).

Cell c₁ is to be placed along the x-axis too. The connectivity is as follows:

• net 1 : p₁ and c₁ (weight: 1)

• net 2 : c₁ and p₂ (weight: 2)

We want to optimize the quadratic wirelength function, which is expressed as (w₁*l(net1))² + (w₂*l(net2))² + ... where w# is the weighting factor for net# and l(net#) is the length of net#. Find the location of c₁ that minimizes the objective function.

2-2. Two pins, p₁ and p₂ are located at a₁ and a₂, respectively, along the x-axis (a₁ < a₂).

Cells c₁ and c₂ are to be placed along the x-axis too. The connectivity is as follows:

• net 1 : p₁ and c₁ (weight: 1)

• net 2 : c₁ and c₂ (weight: 1)

• net 3 : c₂ and p₂ (weight: 2)

We want to optimize the quadratic wirelength function, which is expressed as (w₁*l(net1))² + (w₂*l(net2))² + ... where w# is the weighting factor for net# and l(net#) is the length of net#. Find the locations of c₁ and c₂ that minimize the objective function.